Proposition 2

Tag 0ANN, permanent. Cite it as [Bourbaki, Tag 0ANN].

Lie Groups and Lie Algebras, chapter IV, § 2. Tits Systems

Let X, Y ⊂ S and $w \in W$ . We have

G_X w G_Y = BW_X w W_Y B.

Let $s_1, \ldots, s_q \in X$ and $t_1, \ldots, t_q \in Y$ . Lemma 1 shows that

C(s_1 \ldots s_q).C(w).C(t_1 \ldots t_q) \subset BW_X w W_Y B,

and hence that

G_X w G_Y \subset BW_X w W_Y B.

The opposite inclusion is obvious.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by